304
ANALYTICAL MECHANICS
Expanding the right-hand member of the last equation and rearranging the terms we get
x = (#1 cos <5i + a2 cos <52) sin ut
+ (di sin 81 + a2 sin 52) cos o>t = a cos 5 sin ut + a sin 5 cos ut = a sin (co^ + 5), (3)
where a cos 5 = ai cos 5i + a2 cos 52, and a sin 5 = ai sin 5i + a2 sin <52.
FIG. 134.
It is evident from equation (3) that the resulting motion is simple harmonic and has the same period as the component motions.
Squaring the last two equations and adding we obtain the amplitude of the motion in terms of the constants of equations (1) and (2). Thus
a2 = ai2 + a22 + 2 aia2 cos (52 - Si). (4)
The phase angle of the motion is evidently defined by
n<5i + a2sin52
tan 8 =
cos di + a2 cos
(5)
239. Graphical Method. — The graph of the resulting motion may be obtained by either of the following methods:
(1) Represent the given motions by displacement-time curves, then add the ordinates of these curves in order to